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that calculating the neighborhood with the previous formula
for all j (for all elements of the solution) yielded better results
without significantly increasing algorithm time consumption.
Moreover, according to the author of [14], the lack of
exploitation of the ABC algorithm is partly due to the fact
that the neighborhood function in the original algorithm is
used by both employed and onlooker bees. Thus, no variety
search behavior can be anticipated. Two different equations are
proposed in [14] to calculate the neighborhood. The employed
bees use, instead of Eq. 3, the mutation current-to-rand/1 of
DE algorithm, which is defined as :

Lastly, since the best solution gbest is used to calculate
the neighborhood (for both employed and onlooker phases) as
well as in the mutation phase, it seemed interesting to initialize
gbest with a significant value to accelerate the convergence of
the algorithm and give it a head start. For that, we chose kmeans algorithm to yield the initial value of gbest because of
its simplicity and low time consumption.
Algorithm 2 K-ABC-DE algorithm
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vi = xi (g)+ki ×(xi1 (g)−xi (g))+F 0 ×(xi2 (g)−xi3 (g)) (8)

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where i1 , i2 and i3 are indexes of food sources randomly
chosen so that i1 6= i2 6= i3 6= i. kij is a random value
from a uniform distribution between 0 and 1. F 0 ∈ [0, 1] a
random value from a uniform distribution created once for
each iteration.
As for onlooker bees, the neighborhood equation proposed
in [14] is defined as :
vij = gbestj + F × (xdest,j − xsrc,j )

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(9)

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where gbest denotes the bee with the currently best fitness
value. xdest,j and xsrc,j are the bees chosen randomly such
that f (xdest,j ) < f (xsrc,j ), f being the objective function to
minimize. F here is a randomly chosen value from a uniform
distribution between 0 and 1.
Assuming that the research strategies of the two phases
(employed and onlooker) had to be different and after several
tests, we found that the best combination was to use the
current-to-rand/1 (Eq. 8) in the employed phase and the
GABC formula (Eq. 7) for the onlooker phase.
Again with the aim of improving the ABC algorithm, and
based on a part of the study of D. C. Tran [10], we have
added a phase in each iteration which is the mutation phase.
For that, we used a formula proposed in [10] and inspired by
the mutation of the DE algorithm and the arithmetic crossing
of the HABC [9] algorithm, it is defined as :
uij = Fij0 × (xij − xk1 j ) + Fij × (gbestj − xk2 j )

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(10)
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where xk1 et xk2 are two food sources which are randomly
selected from food source population such that i, k1 and k2
are mutually different. Fij and Fij0 are in this case randomly
chosen between 0 and 1.
One of the advantages of this approach is that it significantly improves the ABC algorithm without increasing the
computation time or adding parameters to be adjusted, since
the mutation operators used do not depend on the parameters
F and F 0 .

Initialize the population solution xi , i = 1, 2, ..., SN ;
Initialize gbest using k-means algorithm (Alg. 1);
Assign employed bees to solutions;
Evaluate the fitness value f it(xi ) of solutions xi
using Eq. 2;
Set triali to 0 , i = 1, 2, ..., SN ;
repeat
/* Employed Phase */
for i = 1 to SN do
Produce a new solution vi from xi using Eq. 8;
Evaluate the fitness value of the new solution
using Eq. 2;
Update triali ;
end for
Calculate the probabilities pi , i = 1, 2, ..., SN
using Eq. 4;
/* Onlooker Phase */
for each onlooker bee do
Select a solution xi depending on pi ;
Produce a new solution vi from xi using Eq. 7;
Evaluate the fitness value of the new solution
using Eq. 2;
Update triali ;
end for
/* Mutation Phase */
for chaque solution xi do
Produce a mutant solution ui using Eq. 10;
Evaluate the fitness value of the new solution
using Eq. 2;
Update triali ;
end for
/* Scout Phase */
for each solution xi do
if triali > limit then
Replace xi with a new solution generated
using Eq. 1;
end if
end for
Memorize the best solution found so far;
until stop condition is satisfied
return gbest;
VI. E XPERIMENTAL R ESULTS AND D ISCUSSION

The proposed algorithm K-ABC-DE is compared with
five other algorithms : ABC, GABC, DE/rand/1/bin (which
uses Eq. 11 as mutation operator and binomial crossover),

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